The maximum stable set problem is one classical combinatorial problem, which aims to find the stability number of a graph. A stable set of a undirected graph \(\mathcal {G}=(\mathcal {V},\mathcal {E})\)  is a set of nodes of \(\mathcal {G}\)  such that there are no edges between them. The maximum stable number of \(\mathcal {G}\) , denoted as \(\alpha (\mathcal {G})\) , is the size of maximum stable set. However, testing whether a \(\alpha (\mathcal {G})\)  is greater than a integer \(k\)  is well-known to be NP-complete [1]}.
This problem can be formulated as
\( \begin{aligned}\alpha (\mathcal {G}) \max _{x_i} \quad & \sum _{i=1}^{n}x_i \\\mathrm {subject~to} \quad & x_i \in \lbrace 0,1\rbrace , \quad i = 1,2,\ldots ,n,\\& x_ix_j = 0, \quad \forall (i,j)\in \mathcal {E}.\end{aligned}\) 
